Twists of Lerch zeta-functions

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1 Twists of Lerch zeta-functions Ramūnas Garunkštis, Jörn Steuding April 2000 Abstract We study twists Lλ, α, s, χ, Q) χn+q)eλn) n+α) of Lerch zeta-functions with s Dirichlet characters χ mod and parameters 0 < α, λ IR, Q Z This generalizes Dirichlet L-functions Lλ, α, s, χ, Q) turns out to be a meromorphic function with atmost one simple pole at s, and satisfies a functional euation similar to the functional euation of the Lerch zetafunction) if χ is primitive Further, we continue our investigations [2] on the zero distributions of these twists We also disproof the analogue of Riemann s hypothesis for L0,, s, χ, Q) when > 3 is prime and Q mod Introduction As usual, let ez) exp2πiz) and denote by {λ} the fractional part of a real number λ, and by m, ) the greatest common divisor of the integers m, For 0 < α, λ IR, the Lerch zeta-function is given by ) Lλ, α, s) eλn) n + α) s This series converges absolutely for Re s > The analytic properties of Lλ, α, s) are uite different depending on λ Z or not If λ Z, then the series ) can be continued analytically to the whole complex plane, and Lλ, α, s) turns out to be an entire function For λ Z the Lerch zeta-function becomes the Hurwitz zeta-function ζs, α) : L, α, s), which can be continued analytically to the whole complex plane except for a simple pole at s with residue Setting λ + {λ} and λ {, λ Z {λ}, λ Z for λ IR, one can prove the functional euation Lλ, α, s) Γs) s 2π) s e 4 αλ ) L α, λ, s) + e s ) 2) 4 + αλ+) Lα, λ +, s)

2 ζs, α) was introduced by Hurwitz [3], Lλ, α, s) by Lipschitz [7] and Lerch [5] independently) Hurwitz and Lerch also gave the first proofs of the corresponding functional euations For 0 < α, λ IR, Q Z we define the twist of the Lerch zeta-function Lλ, α, s) with a Dirichlet character χ mod by Lλ, α, s, χ, Q) χn + Q)eλn) n + α) s for definition and properties of characters see []) Similar to ), this series converges absolutely in Re s > 2 Analytic continuation and functional euation The analytic properties of twisted Lerch zeta-functions depend on the value λ : denote the Gaussian sums to a character χ mod by ) ak τ a χ) : χk)e and τχ) : τ χ) Then a Z) Theorem Let χ be a character mod Then Lλ, α, s, χ, Q) can be continued meromorphically to the whole complex plane with at most one simple pole at s Lλ, α, s, χ, Q) has a pole at s if and only if λ : m Z and τ m χ) 0 In particular, if χ mod is a primitive character we can even prove Theorem 2 Let χ be a primitive character mod If λ is not an integer coprime to, then Lλ, α, s, χ, Q) is an entire function; otherwise, if λ : m Z and m, ), then Lλ, α, s, χ, Q) is a meromorphic function with one simple pole at s with residue τχ)χm)e λq) In any case, Lλ, α, s, χ, Q) satisfies the functional euation ) s Γs) 2π Lλ, α, s, χ, Q) τχ) s χ )e 4 λq α Q +e s 4 λq + α Q ) λ) + L ) λ) L α Q α Q ), λ), s, χ, λ) λ )), λ) +, s, χ, λ) + + λ Setting λ α Q we can deduce from Theorem 2 the well known functional euation of Dirichlet L-functions Ls, χ) : χn) n n to primitive characters χ mod s Remark After preparation of this paper it becomes known to the authors that Bruce C Berndt obtained similar results in Character analogues of the Poisson and Euler-Maclaurin summation formulas with applications, J Number Theory 7 975), , and even more general results in a + 2

3 joint work with Lowell Schoenfeld Periodic analogues of the Euler-Mac Laurin and Poisson summation formulas with applications to number theory, Acta Arithmetica ), Nevertheless, we give now the Proof of Theorem and Theorem 2 Let Re s > Writing Lλ, α, s, χ, Q) in terms of Lerch zeta-functions, we obtain 3) Lλ, α, s, χ, Q) χn + k + Q)eλn + k)) n + k + α) s s χk + Q)eλk)L λ, k + α, s This representation holds by analytic continuation throughout the complex plane By the well known properties of Lerch zeta-functions mentioned above) it follows that Lλ, α, s, χ, Q) is an entire function provided λ Z Otherwise, if λ m Z, then 3) translates to Lλ, α, s, χ, Q) ) ) mk s χk + Q)e ζ s, k + α ) By the properties of Hurwitz zeta-functions see above) Lλ, α, s, χ, Q) turns out to be a meromorphic function with atmost one simple pole at s with principal part 4) s ) mk χk + Q)e s τ mχ)e λq) The principal part of Lλ, α, s, χ, Q) at s vanishes if and only if τ m χ) 0, which proves Theorem Now suppose that χ mod is primitive Recall that the Gaussian sums to primitive characters χ mod satisfy the formulas see []) τχ), 5) τχ) χ )τχ) and τ a χ) τχ)χa) Thus, we deduce from 4), if λ m Z, that the residue of Lλ, α, s, χ, Q) at s is τχ)χm)e λq) This residue vanishes if and only if m is not coprime to Therefore, Lλ, α, s, χ, Q) is regular at s, and hence an entire function, if and only if m, ) > To prove the functional euation we make use of 2) in the representation 3) This yields for Re s > Lλ, α, s, χ, Q) s χk + Q)eλk)L λ, k + α ), s ) s Γs) χk + Q) 2π 3

4 ) e λk + s 4 k + α ) e k+α λ) n n + λ) ) s+ ) +e λk s 4 + k + α ) k+α e λ) + n n + λ) + ) s ) s Γs) 2π ) s e 4 α ) e αn λ) n + λ) ) λ)k n + λ) ) s χk + Q)e + ) +e s 4 + α ) αn e λ)+ n + λ) + ) + λ)k n + λ) + ) s χk + Q)e Once more with 5) we can calculate the occuring character sums by Thus n + λ) ) λ)k χk + Q)e Qn + λ) τχ)χ )χn + λ) ) λ) λ)e, n + λ) + ) + λ)k χk + Q)e τχ)χn + λ) + + λ)e Qn + ) λ)+ + λ) Lλ, α, s, χ, Q) τχ) ) s Γs) 2π s χ )e 4 α ) χn + λ) λ) λ) n + λ) ) s e +e s 4 + α ) λ)+ χn + λ) + + λ) n + λ) + ) s e αn + Qn + λ) λ) αn Qn + λ) + ) ) + λ) Rewriting this in terms of twists of Lerch zeta-functions we obtain the functional euation, which holds by analytic continuation throughout the whole complex plane This proves Theorem 2 If χ is an improper character mod induced by some primitive character χ mod, then 0, ) ),m) τ m χ) m, µ )χ )χ τχ ),,m) ϕ) ϕ,m),m),m),m) ) + 4

5 where µ is the Möbius function and ϕ is Euler s totient This reduces the uestion on the holomorphicity of Lλ, α, s, χ, Q) at s to the arithmetic of χ mod Now suppose that χ mod is primitive By Theorem 2 the values of twisted Lerch zeta-functions at s yield the euivalence ) m L, α,, χ, Q m, ) m Z) τχ) Res sl m, α,, χ, 0 ) χm) defines a homomorphism from Note that the mapping m the group of prime residue classes mod into the group of primitive roots of unity according to the -th cyclotomic polynomial Further, note that one can represent Lerch zeta-functions by twists: since m0 L λ + m ), α, s, χ, Q χn + Q)eλn) n + α) s m0 mn e and the inner sum euals, if n 0 mod, and 0 else, we obtain L λ + m ), α, s, χ, Q s χq)l λ, α ), s m0 This vanishes identically if and only if Q is not coprime to Otherwise, when Q, ), we obtain for 0 < α after λ λ, α α ) Lλ, α, s) s χq) 3 Zero distribution m0 ) λ + m L, α, s, χ, Q We continue our investigations on the zero distributions of Lerch zeta-functions [2], based on a method due to Levinson and Montgomery [6], now on twists to primitive characters As one might expect by comparing the functional euation of Lerch zeta-functions and the one of twists we get very similar results, which we state without proofs As in the case of the Lerch zeta-functions we have to distinguish between so-called trivial and nontrivial zeros The trivial zeros arise more or less directly from the functional euation and lie next to a straight line in the left half of the complex plane; their number inside a circle with radius T around the origin is T, where the implicit constant depends on λ, α, χ, Q and The nontrivial zeros lie in a horizontal bounded strip including the critical strip 0 Re s Denote by NT, λ, α, χ, Q) the number of nontrivial zeros of Lλ, α, s, χ, Q) with Im T according multiplicities) Preceding as in [2] we obtain Theorem 3 Let b 3 be a constant and χ be a primitive character mod Then we have, as T turns to infinity, b + Re ) b + ) T 2 π log T 2πeα λ) + λ) + T 2π log α + Olog T)) λ)+ λ) Im T ), 5

6 Note that the main terms do not depend on χ and Q Using this with b + instead of b, we get after subtracting the resulting formula from the one above Corollary 4 Let χ be a primitive character mod Then we have, as T turns to infinity, NT, λ, α, χ, Q) T π log T 2πeα + Olog T) λ) + λ) Multiplying the formula of Corollary 4 with b gives and subtracting it from the formula of Theorem Corollary 5 Let χ be a primitive character mod Then we have, as T turns to infinity, Re ) T 2 2π log α + Olog T) λ)+ λ) Im T Hence we have Im T Re 2) ont, λ, α, χ, Q)) Moreover, it follows that Lλ, α, s, χ, Q) has infinitely many zeros off the critical line if α 2 λ) + λ) Define 2π Σλ, α) lim T T 0<α,λ Σλ,α)<0 Im T 0 Re ) 2 Then, by Corollary 5, a non-zero Σλ, α) indicates an asymmetrical distribution of the nontrivial zeros of Lλ, α, s, χ, Q) with respect to the critical line) Since π dλdα λ λ)dλ 8, we get Corollary 6 Almost all twists of Lerch zeta-functions Lλ, α, s, χ, Q) to a primitive character have an asymmetrical zero distribution The measure of the twists Lλ, α, s, χ, Q) with negative Σλ, α) is π , and the measure of those Lλ, α, s, χ, Q) with positive Σλ, α) is π The asymmetries vanish if we look at all twists of Lerch zeta-functions to primitive characters: Σλ, α)dλdα 0 0<α,λ 4 Disproof of the analogue of Riemann s hypothesis for certain twists It is expected that Dirichlet L-functions Ls, χ) L0,, s, χ, ) to primitive characters χ mod satisfy the analogue of) Riemann s hypothesis, ie that all nontrivial non-real) zeros lie on the critical line Re s 2 Since Σ0, ) 0 it is impossible to deduce from Corollary 5 that Riemann s hypothesis is false for L0,, s, χ, Q) χn+q ) n n whenever Q mod Nevertheless, we s prove 6

7 Theorem 7 Let > 3 be prime and Q mod Then for any σ, σ 2 with 2 < σ < σ 2 <, there exists a constant c cq, σ, σ 2 ) > 0 such that for sufficiently large T the function L0,, s, χ, Q) has more than ct zeros in the rectangle σ < σ < σ 2, t < T Note, that this theorem is true for any not necessarily primitive) character χ We say that two characters are euivalent if they are generated by the same primitive character Further, denote by meas{a} the Lebesgue measure of the set A The proof of Theorem 7 makes use of Voronin s theorem on the universality of Dirichlet L-functions: Lemma 8 Voronin [8]) Let 0 < r < 4 Let χ,, χ n be pairwise not euivalent Dirichlet characters Let f s),, f n s) be any nonvanishing continuous functions on the disc s r, analytic in the interior of this disc Then for every ε > 0 lim inf T 2T meas { τ [ T, T], max max j n s r s L + 3 ) } 4 + iτ, χ j f j s) < ε > 0 Further, we need some information on certain sums of characters: let χ be an arbitrary Dirichlet character mod and denote by χ n, n,, the characters mod, where is prime Define 6) η n Q) χk + Q) χ n k + ) n,, ) Lemma 9 Let > 3 be prime and Q mod Then there exist at least two different nonprincipal characters, say χ i and χ j, such that the numbers η i Q) and η j Q) are non-zero Proof of Lemma 9 By the orthogonality relation of Dirichlet characters we obtain ϕ) η n Q) 2 n k,l0 k+q,) ϕ) χk + ) χl + ) 2) ) n χk + ) χk + ) ) χ n k + Q) χ n l + Q) First, suppose that χ is a non-principal character mod By χ we denote the principal character mod Then η Q) k+,) χk + Q) χk + Q) χ + Q) χq ) 0 7

8 If we suppose that only two numbers η Q) and, say, η i Q), i, are non-zero, then ϕ) 2) ) η n Q) 2 + η i Q) 2 n This is impossible since η i Q) 2) This proves the lemma when χ is a non-principal character If χ is principal, then η Q) 2 and η n Q) for n 2,, Proof of Theorem 7 following some ideas of Voronin [8] and Laurinčikas [4]) For k IN with k < we have ζ s, k ) s χ n k)ls, χ n ) n In view of this, 3) and 6) we get L0,, s, χ, Q) s χk + Q )ζs, k ) + χq )ζs) s k η n Q)Ls, χ n ) + sχq )ζs) n If χ denotes the principal character mod we have ζs) 7) L0,, s, χ, Q) η n Q)Ls, χ n ) + s ) Ls, χ ) Thus η Q) χq ) + s ) ) s Ls, χ ) η n Q)Ls, χ n ) + η Q) + ) s χq ) Ls, χ ) By Lemma 9 we can suppose that η 2 Q) and η 3 Q) are non-zero Let σ 2 σ + σ 2 ) and f 2 s) f 3 s) η2 Q)s σ) + 3), 3η3 Q), f s) f 4 s) f ) δ > 0, where the constant δ will be chosen later From Lemma 8 it follows that there exists c c, σ, σ 2 ) > 0 such that for sufficiently large T { } 8) meas τ [ T, T], max k max s 4 max σ 3 4, σ ) Ls + iτ, χ k) f k s) < δ > ct If τ is from 8) and Re s 2, then 9) s+iτ ) Ls + iτ, χ ) s ) f s) s+iτ ) Ls + iτ, χ ) s+iτ ) f s) + f s) s+iτ ) s ) < 3δ 8

9 From 8), 9) and the ineuality η n Q) 2 we deduce that η n Q)Ls + iτ, χ n ) + η Q) + ) 0) s+iτ χq ) Ls + iτ, χ )+ η n Q)f n s) + η Q) + ) s χq ) f s) < 2 δ From the definition of the functions f n s) it follows that η n Q)f n s) + η Q) + ) s χq ) f s) s σ) + R, where R < 2 δ From this and 0), choosing δ < σ2 σ 20, we obtain 2 max η n Q)Ls + iτ, χ n ) + η Q) + ) s+iτ χq ) Ls + iτ, χ ) s σ) s σ 2 σ2 σ) < 2 2 δ < 0 σ 2 σ ) The last ineuality shows, that in the disc s σ 2 σ 2 σ ) the functions s σ and η n Q)Ls + iτ, χ n ) + η Q) + ) s+iτ χq ) Ls + iτ, χ ) s σ) satisfy the conditions of Rouche s theorem Since the function s σ has a zero inside s σ 2 σ 2 σ ), the function η n Q)Ls + iτ, χ n ) + η Q) + ) s+iτ χq ) Ls + iτ, χ ) also has a zero in the same disc By 8) the measure of such τ [ T, T] is greater than ct, which proves the theorem References [] T Apostol, Introduction to Analytic Number Theory, Springer 976 [2] R Garunkštis, J Steuding, On the zero distributions of Lerch zeta-functions, submitted for publication [3] A Hurwitz, Einige Eigenschaften der Dirichletschen Funktionen Fs) D n ) n s, die bei der Bestimmung der Klassenzahlen binärer uadratischer Formen auftreten, Zeitschrift f Math u Physik ),

10 [4] A Laurinčikas, On zeros of some Dirihlet series, Liet Matem Rink 244) 984), 6-26 Russian) [5] M Lerch, Note sur la fonction Kw, x, s) e2kπix w + k) s, Acta Math 887), 9-24 [6] N Levinson, HL Montgomery, Zeros of the derivative of the Riemann zet-function, Acta Math ), [7] R Lipschitz, Untersuchung einer aus vier Elementen gebildeten Reihe, J reine und angew Math ), [8] S M Voronin, Analytical properties of the Dirichlet generating series of arithmetical objects, Doctoral thesis, Moskow, 977) Russian) Ramūnas Garunkštis Department of Mathematics and Informatics Vilnius University Naugarduko Vilnius Lithuania ramunasgarunkstis@mafvult Jörn Steuding Mathematisches Seminar Johann Wolfgang Goethe-Universität Frankfurt Robert-Mayer-Str 0 D Frankfurt Germany steuding@mathuni-frankfurtde 0